Quotient Kinematics Machines: Concept, Analysis, and Synthesis

Abstract

This paper presents a geometric analysis and synthesis theory for quotient kinematics machines (QKMs). Given a desired motion type described by a subgroup G of the special Euclidean group SE(3), QKM refers to a left-and-right hand system that realizes G through the coordinated motion of two mechanism modules, one synthesizing a subgroup H of G, and the other a complement of H in G, denoted by G/H. In the past, QKMs were often categorized into hybrid kinematics machines (HKMs) and were treated on a case-by-case basis. We show that QKMs do have a unique and well-defined kinematic structure that permits a unified and systematic treatment of their synthesis and design. We also study the properties of G/H as a novel motion type for parallel kinematics machine (PKM) synthesis. Another contribution of the paper is to model five-axis machines by SE(3)/R(o,z) (where R(o,z) represents the spindle symmetry) and give a complete classification of five-axis QKMs using the same geometric framework.

Equivalence relation and partial ordering of parallel mechanisms with applications to mechanism synthesis and mobility analysis

“…This subsection discusses applications of partial order and equivalence relation to the analysis and synthesis theory of quotient mechanisms [18]. Take the Echospeed FHT machine center of DS Technology for example, the Sprint Z3 parallel module (3-PRS mechanism) has the motion type…”

“…ate for the definition of 5-axis machining task. Based on the notion of quotient motion type, this paper proposed a synthesis theory [18] for quotient mechanisms. A quotient mechanism consists of two independent motion modules, which may be either a serial mechanism or a parallel one, acting in unison to accomplish one overall motion task.…”

Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics

“…Not all motion patterns can be modelled by Lie subgroups of SE (3). The motion patterns of two DoF robot wrists or orientation devices ( [25]- [27]), three to five DoF haptic devices ( [28]), and five-axis machines ( [29,30]) for example, can only be modelled by submanifolds of SE (3). General submanifolds of SE(3) lack group structure and Lie algebraic properties, and in general defy a systematic classification.…”

Analysis and Synthesis of a Kind of Mobility Reconfigurable Robot with Multi-Task Capability

“…The special Euclidean group (3) SE , which is defined as a closed subgroup of the general linear group ( ) { } with matrix multiplication as the group operation, together with its subgroups and submanifolds play an important role in kinematic analysis and synthesis of the robotic mechanisms [16][17][18] [19] . If the homogeneous representations for points in 3 R Euclidean space are used, the rigid displacement set of a rigid body (represented by a Cartesian frame T attached to the rigid body) relative to a reference configuration (represented by a fixed Cartesian coordinate frame F in Euclidean space and it is assumed that T initially coincides with F ) can be identified with an element ( )…”

“…, the CKM becomes a quotient kinematic structure [19] , which is described as a subgroup task motion { } Fig. 7(d).…”